Assignment 3: Field Lines and Dipoles

Due: 8:00am on Wednesday, January 18, 2012

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Visualizing Electric Fields

Learning Goal: To understand the nature of electric fields and how to draw field lines.

Electric field lines are a tool used to visualize electric fields. A field line is drawn beginning at a positive charge

and ending at a negative charge. Field lines may also appear from the edge of a picture or disappear at the edge of

the picture. Such lines are said to begin or end at infinity. The field lines are directed so that the electric field at

any point is tangent to the field line at that point.

The figure shows two different ways to visualize an electric

field. On the left, vectors are drawn at various points to show the direction and magnitude of the electric field. On

the right, electric field lines depict the same situation. Notice that, as stated above, the electric field lines are drawn

such that their tangents point in the same direction as the electric field vectors on the left. Because of the nature of

electric fields, field lines never cross. Also, the vectors shrink as you move away from the charge, and the electric

field lines spread out as you move away from the charge. The spacing between electric field lines indicates the

strength of the electric field, just as the length of vectors indicates the strength of the electric field. The greater the

spacing between field lines, the weaker the electric field. Although the advantage of field lines over field vectors

may not be apparent in the case of a single charge, electric field lines present a much less cluttered and more

intuitive picture of more complicated charge arrangements.

Part A

Which of the following figures correctly depicts the field lines

from an infinite uniformly negatively charged sheet? Note that the sheet is being viewed edge-on in all pictures.

Hint A.1 Description of the field

Hint not displayed

ANSWER:

A

B

C

D

Correct

Part B

In the diagram from part A , what is wrong with figure B?

(Pick only those statements that apply to figure B.)

Check all that apply.

ANSWER:

Field lines cannot cross each other.

The field lines should be parallel because of the sheet's symmetry.

The field lines should spread apart as they leave the sheet to indicate the weakening of the

field with distance.

The field lines should always end on negative charges or at infinity.

Correct

Part C

Which of the following figures shows the correct electric field

lines for an electric dipole?

ANSWER:

A

B

C

D

Correct

This applet shows two charges. You can alter the charge on each independently or alter the distance between

them. You should try to get a feeling for how altering the charges or the distance affects the field lines.

Part D

In the diagram from part C , what is wrong with figure D?

(Pick only those statements that apply to figure D.)

Check all that apply.

ANSWER:

Field lines cannot cross each other.

The field lines should turn sharply as you move from one charge to the other.

The field lines should be smooth curves.

The field lines should always end on negative charges or at infinity.

Correct

In even relatively simple setups as in the figure, electric field

lines are quite helpful for understanding the field qualitatively (understanding the general direction in which a

certain charge will move from a specific position, identifying locations where the field is roughly zero or where

the field points a specific direction, etc.). A good figure with electric field lines can help you to organize your

thoughts as well as check your calculations to see whether they make sense.

Part E

In the figure , the electric field lines are shown for a system of

two point charges, and . Which of the following could represent the magnitudes and signs of and ?

In the following, take to be a positive quantity.

ANSWER:

,

,

,

,

,

Correct

Very far from the two charges, the system looks like a single charge with value . At large enough

distances, the field lines will be indistinguishable from the field lines due to a single point charge .

Torque on a Dipole in a Uniform Field

Consider an electric dipole whose dipole moment (a vector

pointing from the negitive charge to the positive charge) is oriented at angle with respect to the y axis. There is

an external electric field of magnitude (independent of the field produced by the dipole) pointing in the positive

y direction. The positive and negative ends of the dipole have charges and , respectively, and the two

charges are a distance apart. The dipole has a moment of inertia about its center of mass. It will help you to

imagine that the dipole is free to rotate about a pivot through its center.

Part A

What is the net force that the dipole experiences due to the electric field?

Hint A.1 What is the force on the positive charge?

Hint not displayed

Express in terms of the given variables and the unit vectors , .

ANSWER:

=

Correct

Part B

What is , the magnitude of the torque that the electric field exerts about the center of mass of the dipole?

Hint B.1 Find the torque on the positive charge

Hint not displayed

Hint B.2 Now consider the negative charge

Hint not displayed

Express the magnitude of the total torque in terms of the given quantities.

ANSWER:

=

Correct

Part C

Using the above result, find the potential energy associated with the dipole's orientation in the field as a

function of the angle shown in the figure. Take the zero of the potential to occur when the dipole is at angle

; that is, .

Hint C.1 The definition of potential energy

Hint not displayed

Hint C.2 How to choose the constant of integration

Hint not displayed

Express in terms of the given quantities.

ANSWER:

=

Correct

The dipole moment of a neighboring pair of opposite charges of equal magnitude is defined to be a vector from

negative to positive charge of magnitude equal to the product of and the distance between the charges.

Part D

In terms of the dipole moment , which of the following is an expression for the torque on a dipole in the field

?

Hint D.1 How to approach this part

Hint not displayed

Hint D.2 Formula for

Hint not displayed

Hint D.3 Formula for

Hint not displayed

Hint D.4 More on torques

Hint not displayed

ANSWER:

Correct

Part E

In terms of the dipole moment , which of the following is an expression for the potential energy of a dipole in

the field ?

Hint E.1 Formula for

Hint not displayed

Hint E.2 Formula for

Hint not displayed

ANSWER:

Correct

Electric Dipole in an Electric Field

Point charges 4.00 and 4.00 are separated by distance 3.60 , forming an electric dipole.

Part A

Find the magnitude of the electric dipole moment.

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Formula for dipole moment

Hint not displayed

Express your answer in coulomb meters to three significant figures.

ANSWER:

1.441011

Correct

Part B

What is the direction of the electric dipole moment?

ANSWER:

from to

from to

Correct

Part C

The charges are in a uniform electric field whose direction makes an angle 36.9 with the line connecting the

charges. What is the magnitude of this field if the torque exerted on the dipole has magnitude 7.40109

?

Hint C.1 Find an equation for the torque

Hint not displayed

Hint C.2 How to obtain

Hint not displayed

Express your answer in newtons per coulomb to three significant figures.

ANSWER:

856

Correct

Electrostatic Force of Water on a Chlorine Ion

The dipole moment of the water molecule is . Consider a water molecule located at the

origin whose dipole moment points in the positive x direction. A chlorine ion , of charge ,

is located at meters. Assume that this x value is much larger than the separation between the

charges in the dipole, so that the approximate expression for the electric field along the dipole axis can be used.

Part A

Find the magnitude of the electric force, ignoring the sign, that the water molecule exerts on the chlorine ion.

Hint A.1 How to approach the problem

Hint not displayed

ANSWER:

6.581013

Correct

Part B

What is the direction of the electric force?

ANSWER:

negative x

positive x

Correct

Since the dipole moment points in the positive x direction, the positive side of the dipole is to the right of the

negative side of the dipole. The electric field along the x axis due to the dipole will then point to the right

everywhere (except the small interval between the two charges of the dipole). Try to visualize the electric field of

a dipole if you need clarification, and think about which way the electric field lines point on each end (left and

right). Since the field will point to the right at the location of the chlorine ion, and since the ion is negatively

charged, the force exerted on the ion will be to the left, or along the negative x direction.

Part C

Is this force attractive or repulsive?

ANSWER:

attractive

repulsive

Correct

Since the ion is along the positive x axis, and the force is pointing in the negative x direction, the force is pulling

the chlorine ion toward the dipole (which is at the origin), and so the force is attractive.

Misprints in Exercise 21:60-- Refer to Example 21.14 and Figure 21.33

Exercise 21.60

Consider the electric dipole of Example 21.15.

Part A

Derive an expression for the magnitude of the electric field produced by the dipole at a point on the -axis in

Fig.21.34 in the textbook.

Express your answer in terms of the variables , , , and appropriate constants.

ANSWER:

= Correct

Part B

What is the direction of this electric field?

ANSWER:

= 270

Correct counterclockwise from the -direction

Part C

How does the electric field at points on the -axis depend on when is very large?

Express your answer in terms of the variables , , , and appropriate constants.

ANSWER:

= Correct

Dipole Motion in a Uniform Field

Consider an electric dipole located in a region with an electric

field of magnitude pointing in the positive y direction. The positive and negative ends of the dipole have charges

and , respectively, and the two charges are a distance apart. The dipole has moment of inertia about its

center of mass. The dipole is released from angle , and it is allowed to rotate freely.

Part A

What is , the magnitude of the dipole's angular velocity when it is pointing along the y axis?

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Find the potential energy

Hint not displayed

Hint A.3 Find the total energy at the moment of release

Hint not displayed

Hint A.4 Find the total energy when

Hint not displayed

Express your answer in terms of quantities given in the problem introduction.

ANSWER:

=

Correct

Thus increases with increasing , as you would expect. An easier way to see this is to use the trigonometric

identity

to write as .

Part B

If is small, the dipole will exhibit simple harmonic motion after it is released. What is the period of the

dipole's oscillations in this case?

Hint B.1 How to approach the problem

Hint not displayed

Hint B.2 Compute the torque

Hint not displayed

Hint B.3 The small-angle approximation

Hint not displayed

Hint B.4 Find the oscillation frequency

Hint not displayed

Hint B.5 The relationship between (angular) oscillation frequency and period

Hint not displayed

Express your answer in terms of and quantities given in the problem introduction.

ANSWER:

= Correct